What is a differential equation?

The idea

An algebraic equation says "this equals that." A differential equation says "this rate of change equals that." In symbols: instead of solving for a number, you solve for a function y(t), and the equation involves y, dy/dt, d²y/dt², or higher derivatives.

If you know how fast a population is growing at every moment — that's a differential equation. If you know how fast something cools, how a spring stretches, how a charge flows through a wire — all differential equations.

Order and initial conditions

The order is the highest derivative that appears. dy/dt = ky is first order; d²y/dt² = −ω²y is second order. A first-order ODE typically needs one initial condition (the value of y at t = 0); a second-order ODE typically needs two.

Initial conditions pin down which of the equation's many solutions you actually mean — without them, the answer is a whole family.

Worked example

Solve dy/dt = 2y with y(0) = 5.

dy / y = 2 dt ∫ (1/y) dy = ∫ 2 dt ln |y| = 2t + C y(t) = A · e^(2t) y(0) = A = 5 ⟹ y(t) = 5 e^(2t)

The solution is exponential growth at rate 2. That isn't a guess — it's the unique function whose rate of change equals 2 times itself.

Check your understanding

What is the order of d³y/dt³ + 4 d²y/dt² + y = sin(t)?

First Second Fourth Third